Question

A shell of mass m moving with V suddenly breaks into two pieces. The part having $\dfrac{m}{4}$ remains stationary. The velocity of the other shall will be
A. $V$
B. $2V$
C. $\dfrac{3}{4}V$
D. $\dfrac{4}{3}V$

Answer 1

By using conservation of linear momentum we can solve this problem. In this problem we know the mass of the shell, we also know the initial velocity of the shell and at the same time we know the final velocity of one part of the shell. Now with the help of the momentum formula we can calculate the final velocity of the other shell.

Complete step by step answer:
As per the given problem,$V$ is the initial velocity of the shell when the mass of the shell is $m$. While moving at the $V$ it suddenly breaks into two pieces. ${m_1}$ is the mass of part one and ${m_2}$ is the mass of the second part.As given in the problem,
${m_1} = \dfrac{m}{4}$
If the total mass of the shell is m then,
$m = {m_1} + {m_2}$
By putting the know values we can write,
$m = \dfrac{m}{4} + {m_2}$
$\Rightarrow {m_2} = m - \dfrac{m}{4}$
$\Rightarrow {m_2} = \dfrac{{4m - m}}{4}$
$\Rightarrow {m_2} = \dfrac{{3m}}{4}$
Hence we get to know the mass of the second part that is $\dfrac{{3m}}{4}$ .

Now applying the conservation of momentum formula we can calculate the velocity of the second part. Now,
${P_{initial}} = {P_{final}}$
Where, ${P_{\operatorname{in} ital}} =$ Initial momentum of the shell and ${P_{final}} =$ Final momentum of the shell.
Now putting the value of initial and final momentum in the conservational formula we get,
$m \times V = {m_1} \times {V_1} + {m_2} \times {V_2}$
Now putting the value of ${m_1},{m_2}$ and ${V_1}$ ,we get
$mV = \dfrac{m}{4} \times 0 + \dfrac{{3m}}{4} \times {V_2}$
$\Rightarrow mV = \dfrac{{3m}}{4}{V_2}$
Cancelling the common terms we get,
$V = \dfrac{3}{4}{V_2}$
Rearranging the term we get,
$\therefore {V_2} = \dfrac{4}{3}V$
Hence the velocity of the second part is $\dfrac{4}{3}V$ .

Therefore the correct option is $\dfrac{4}{3}V$ that is $\left( D \right)$.

Note: Generally we get confused in this type of problem because the mass of the other body is not given how we will calculate its velocity. But the same shell is only divided into two parts hence we can calculate the other part mass as the total mass of the shell is equal to the sum of two separated parts.